Nicolas Merener

Numerical solution of jump-diffusion LIBOR market models

Abstract. This paper develops, analyzes, and tests computational procedures for the numerical solution of LIBOR market models with jumps. We consider, in particular, a class of models in which jumps are driven by marked point processes with intensities that depend on the LIBOR rates themselves. While this formulation offers some attractive modeling features, it presents a challenge for computational work. As a first step, we therefore show how to reformulate a term structure model driven by marked point processes with suitably bounded state-dependent intensities into one driven by a Poisson random measure. This facilitates the development of discretization schemes because the Poisson random measure can be simulated without discretization error. Jumps in LIBOR rates are then thinned from the Poisson random measure using state-dependent thinning probabilities. Because of discontinuities inherent to the thinning process, this procedure falls outside the scope of existing convergence results; we provide some theoretical support for our method through a result establishing first and second order convergence of schemes that accommodates thinning but imposes stronger conditions on other problem data. The bias and computational efficiency of various schemes are compared through numerical experiments.

Convergence of a discretization scheme for jump-diffusion processes with state–dependent intensities

This paper proves a convergence result for a discretization scheme for simulating jump–diffusion processes with state–dependent jump intensities. With a bound on the intensity, the point process of jump times can be constructed by thinning a Poisson random measure using state–dependent thinning probabilities. Between the jump epochs of the Poisson random measure, the dynamics of the constructed process are purely diffusive and may be simulated using standard discretization methods. Under conditions on the coefficient functions of the jump–diffusion process, we show that the weak convergence order of this method equals the weak convergence order of the scheme used for the purely diffusive intervals: the construction of jumps does not degrade the convergence of the method.

Globally Distributed Production and the Pricing of CME Commodity Futures

I investigate how local supply shocks in the globally distributed production of commodities are incorporated into Chicago Mercantile Exchange (CME) futures prices. I exploit that the soybean market share of the United States (Argentina) decreased (increased) between 1996 and 2010, and use rain, which tends to increase output, as a source of exogenous supply shocks. I find a significantly negative response of CME soybean prices to daily rain across regions and time. Moreover, the impact of local rain on the CME price is approximately linear in the time‐varying local share of global output. Therefore, traders of CME contracts seem to aggregate supply in a globally integrated manner and are exposed to globally distributed shocks.

Swap Rate Variance Swaps

We study the hedging and valuation of generalized variance swaps de¯ned on a forward swap interest rate. Our motivation is the fundamental role of variance swaps in the transfer of variance risk, and the extensive empirical evidence documenting that the variance realized by interest rates is stochastic. We identify a hedging rule involving a static European contract and the gains of a dynamic position on forward interest rate swaps. Two distinguishing features arise in the context of interest rates: the nonlinear and multidimensional relationship between the values of the dynamically traded contracts and the underlying swap rate, and the possible stochasticity of the interest rate at which gains are reinvested. The combination of these two features leads to additional terms in the cumulative dynamic trading gains, which depend on realized variance and are taken into consideration in the determination of the appropriate static hedge. We characterize the static payo® function as the solution of an ordinary di®erential equation, and derive explicitly the associated dynamic strategy. We use daily interest rate data between 1997 and 2007 to test the e®ectiveness of our hedging methodology in arithmetic and geometric variance swaps and verify that the hedging error is small compared to the bid-ask spread in swaption prices.

An Efficient Monte Carlo Method for Discrete Variance Contracts

We develop an efficient Monte Carlo method for the valuation of financial contracts on discretely realized variance. We work with a general stochastic volatility model that makes realized variance dependent on the full path of the asset price. The variance contract price is a high-dimensional integral over the fundamental sources of randomness. We identify a two-dimensional manifold that drives most of the uncertainty in realized variance, and we compute the contract price by combining precise integration over this manifold, implemented as fine stratification or deterministic sampling with quasirandom numbers, with conditional Monte Carlo on the remaining dimensions. For a subclass of models and a class of nonlinear payoffs, we derive approximate theoretical results that quantify the variance reduction achieved by our method. Numerical tests for the discretized versions of the widely used Hull–White and Heston models show that the algorithm performs significantly better than a standard Monte Carlo, even for fixed computational budgets.

Optimal Trading and Shipping of Agricultural Commodities

We develop and implement a model for a profit maximizing firm that provides an intermediation service between commodity producers and commodity end-users. We are motivated by the grain intermediation business at Los Grobo—one of the largest commodity-trading firms in South America. Producers and end-users are distributed over a realistic spatial network, and trade with the firm through contracts for delivery of grain during the marketing season. The firm owns spatially distributed storage facilities, and begins the marketing season with a portfolio of prearranged purchase and sale contracts with upstream and downstream counterparts. The firm aims to maximize profits while satisfying all previous commitments, possibly through the execution of new transactions. Under realistic constraints for capacities, network structure and shipping costs, we identify the optimal trading, storing and shipping policy for the firm as the solution of a profit-maximizing optimization problem, encoded as a minimum cost flow problem in a time-expanded network that captures both geography and time. We perform extensive numerical examples and show significant efficiency gains derived from the joint planning of logistics and trading.

Concentrated Production and Heavy Tails in Commodity Returns

I study the impact of commodity production concentration on the occurrence of extreme commodity returns. I explore this issue in a sample of 22 agricultural, mineral and energy commodities of global scope that are liquidly traded through futures at the most important exchanges. I find that measures of production concentration such as the Herfindahl index computed on national shares of global output, or the market share of the top three producers, had significant and positive effect on measures of extreme returns during 1995-2012, as implied by daily return kurtosis or the shape parameter of the distribution of extreme returns. Volatility persistence appears unlikely to generate the cross sectional kurtosis observed empirically unless heavy tailed conditional returns are also included in the dynamics. The results are economically significant and robust to the inclusion of controls for inventories, futures liquidity and country size. These findings are consistent with a simple mechanism of aggregation of globally distributed local supply shocks that impact global supply and hence commodity prices.

Efficient Monte Carlo for Discrete Variance Contracts

We develop an efficient Monte Carlo method for the valuation of a financial contract with payoff dependent on discretely realized variance. We assume a general model in which asset returns are random shocks modulated by a stochastic volatility process. Realized variance is the sum of squared daily returns, depending on the sequence of shocks to the asset and the realized path of the volatility process. The price of interest is the expected payoff, represented as a high dimensional integral over the fundamental sources of randomness. We compute it through the combination of deterministic integration over a two dimensional manifold defined by the sum of squared shocks to the asset and the path average of the modulating variance process, followedby exact conditional Monte Carlo sampling. The deterministic integration variables capture most of the variability in realized variance therefore the residual variance in our estimator is much smaller than that in standard Monte Carlo. We derive theoretical results that quantify the variance reduction achieved by the method. We test it for the Hull-White, Heston, and Double Exponential models and show that the algorithm performs significantly better than standard Monte Carlo for realistic computational budgets.